Electron orbital models

However, in the case of the electronic orbital model there is no way in which the inter-electronic repulsions can be physically reduced. This form of distinction has not been sufficiently emphasized by philosophers. I believe that the nature of the orbital model shows that not all theoretical models can be lumped together as in the work of Achinstein [1968]. 

Electrons rotate around the nucleus at high speeds, and determining their exact location at any given time is impossible. Nevertheless, for simplicity, electrons often are depicted as small spheres occupying discrete orbits (fig. 1). A more realistic depiction, the electron orbital model, indi-... 

Electron orbitals. Model depicting the volume of space in which an electron is likely to be found 90% of the time, (a) The first energy level consists of one spherical orbital containing up to two electrons. The second energy level has four orbitals, each describing the distribution of up to two electrons. One of the orbitals of the second energy level is spherical the other three are dumbbell-shaped and arranged perpendicular to one another, as the axis lines indicate (b). The nucleus is at the center, where the axes intersect. 

In the Bom-Oppenlieimer [1] model, it is assumed that the electrons move so quickly that they can adjust their motions essentially instantaneously with respect to any movements of the heavier and slower atomic nuclei. In typical molecules, the valence electrons orbit about the nuclei about once every 10 s (the iimer-shell electrons move even faster), while the bonds vibrate every 10 s, and the molecule rotates... 

So, within the limitations of the single-detenninant, frozen-orbital model, the ionization potentials (IPs) and electron affinities (EAs) are given as the negative of the occupied and virtual spin-orbital energies, respectively. This statement is referred to as Koopmans theorem [47] it is used extensively in quantum chemical calculations as a means for estimating IPs and EAs and often yields results drat are qualitatively correct (i.e., 0.5 eV). 

Figure Cl. 1.2. (a) Mass spectmm of sodium clusters (Na ), N= 4-75. The inset corresponds to A = 75-100. Note tire more abundant clusters at A = 8, 20, 40, 58, and 92. (b) Calculated relative electronic stability, A(A + 1) - A(A0 versus N using tire spherical electron shell model. The closed shell orbitals are labelled, which correspond to tire more abundant clusters observed in tire mass spectmm. Knight W D, Clemenger K, de Heer W A, Saunders W A, Chou M Y and Cohen ML 1984 Phys. Rev. Lett. 52 2141, figure 1.

VV e now wish to establish the general functional form of possible wavefunctions for the two electrons in this pseudo helium atom. We will do so by considering first the spatial part of the u a efunction. We will show how to derive functional forms for the wavefunction in which the i change of electrons is independent of the electron labels and does not affect the electron density. The simplest approach is to assume that each wavefunction for the helium atom is the product of the individual one-electron solutions. As we have just seen, this implies that the total energy is equal to the sum of the one-electron orbital energies, which is not correct as ii ignores electron-electron repulsion. Nevertheless, it is a useful illustrative model. The wavefunction of the lowest energy state then has each of the two electrons in a Is orbital ... 

We 11 expand our picture of bonding by introducing two approaches that grew out of the idea that electrons can be described as waves—the valence bond and molecular orbital models In particular one aspect of the valence bond model called orbital hybridization, will be emphasized... 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

FIGURE 4 19 Bonding in methyl radical (a) If the structure of the CH3 radical IS planar then carbon is sp hybridized with an unpaired electron in 2p orbital (b) If CH3 IS pyramidal then car bon IS sp hybridized with an electron in sp orbital Model (a) IS more consistent with experimental observa tions... 

Carbon has six electrons around the atomic core as shown in Fig. 2. Among them two electrons are in the K-shell being the closest position from the centre of atom, and the residual four electrons in the L-shell. TTie former is the Is state and the latter are divided into two states, 2s and 2p. The chemical bonding between neighbouring carbon atoms is undertaken by the L-shell electrons. Three types of chemical bonds in carbon are single bond contributed from one 2s electron and three 2p electrons to be cited as sp bonding, double bond as sp and triple bond as sp from the hybridised atomic-orbital model. 

The orbital model would be exact were the electron repulsion terms negligible or equal to a constant. Even if they were negligible, we would have to solve an electronic Schrodinger equation appropriate to CioHs " " in order to make progress with the solution of the electronic Schrodinger equation for naphthalene. Every molecular problem would be different. 

There are m doubly occupied molecular orbitals, and the number of electrons is 2m because we have allocated an a and a spin electron to each. In the original Hartree model, the many-electron wavefunction was written as a straightforward product of one-electron orbitals i/p, i/ and so on... 

Atoms are special, because of their high symmetry. How do we proceed to molecules The orbital model dominates chemistry, and at the heart of the orbital model is the HF-LCAO procedure. The main problem is integral evaluation. Even in simple HF-LCAO calculations we have to evaluate a large number of integrals in order to construct the HF Hamiltonian matrix, especially the notorious two-electron integrals... 

In order to retain the orbital model for a many-electron atom, Hartree assumed that each electron came under the influence of the nuclear charge and an average potential due to the remaining electrons. He therefore retained the form of the radial equation for a one-electron atom, equation 12.2, but assumed that the mutual potential energy U was the sum of... 

In Chapter 9, we considered a simple picture of metallic bonding, the electron-sea model The molecular orbital approach leads to a refinement of this model known as band theory. Here, a crystal of a metal is considered to be one huge molecule. Valence electrons of the metal are fed into delocalized molecular orbitals, formed in the usual way from atomic... 

Paper four first appeared in the Journal of Chemical Education and aimed to highlight one of the important ways in which the periodic table is not fully explained by quantum mechanics. The orbital model and the four quantum number description of electrons, as described earlier, is generally taken as the explanation of the periodic table but there is an important and often neglected limitation in this explanation. This is the fact that the possible combinations of four quantum numbers, which are strictly deduced from the theory, explain the closing of electron shells but not the closing of the periods. That is to say the deductive explanation only shows why successive electron shells can contain 2, 8, 18 and 32 electrons respectively. 

This paper deals with some questions in the foundations of chemistry. The atomic orbital (or electronic configuration) model is examined, with regards to both its origins and current usage. I explore the question of whether the commonly-used electronic configuration of atoms have any basis in quantum mechanics as is often claimed particularly in chemical education. 

It is now known that the view of electrons in individual well-defined quantum states represents an approximation. The new quantum mechanics formulated in 1926 shows unambiguously that this model is strictly incorrect. The field of chemistry continues to adhere to the model, however. Pauli s scheme and the view that each electron is in a stationary state are the basis of the current approach to chemistry teaching and the electronic account of the periodic table. The fact that Pauli unwittingly contributed to the retention of the orbital model, albeit in modified form, is somewhat paradoxical in view of his frequent criticism of the older Bohr orbits model. For example Pauli writes,... 

In spite of the insecure basis which the orbital model possesses, it has proved fruitful in the field of atomic chemistry and physics. Firstly the use of electronic configurations serve as a basis for the classification of the lines shown by atomic spectra (Condon and Shortley [1935], Slater [1949]). 

In 1926 Llewellyn Thomas proposed treating the electrons in an atom by analogy to a statistical gas of particles. Electron-shells are not envisaged in this model, which was independently rediscovered by Enrico Fermi two years later. For many years the Thomas-Fermi method was regarded as a mathematical curiosity without much hope of application since the results it yielded were inferior to those obtained by the method based on electron orbitals.17... 

As described in the opening pages of Chapter 9, the electrons in a hydrogen molecule are smeared out between the two nuclei in a way that maximizes electron/nucleus attraction. To understand chemical bonding, we must develop a new orbital model that accounts for shared electrons. In other words, we need to develop a set of bonding orbitals. 

Our treatment of O2 shows that the extra complexity of the molecular orbital approach explains features that a simpler description of bonding cannot explain. The Lewis structure of O2 does not reveal its two unpaired electrons, but an MO approach does. The simple (t-tt description of the double bond in O2 does not predict that the bond in 2 is stronger than that in O2, but an MO approach does. As we show in the following sections, the molecular orbital model has even greater advantages in explaining bonding when Lewis structures show the presence of resonance. 

Let us summarize briefly at this stage. We have seen that the point of degeneracy forms an extended hyperline which we have illnstrated in detail for a four electrons in four Is orbitals model. The geometries that lie on the hyperline are predictable for the 4 orbital 4 electron case using the VB bond energy (Eq. 9.1) and the London formula (Eq. 9.2). This concept can be nsed to provide nseful qualitative information in other problems. Thns we were able to rationalize the conical intersection geometry for a [2+2] photochemical cycloaddition and the di-Jt-methane rearrangement. 

---